A COUPLED ELECTROMAGNETIC-THERMAL MODEL
OF HEATING DURING RADIOFREQUENCY ABLATION
A Thesis
Presented in Partial Fulfillment of the Requirements for
the Degree Master of Science in the
Graduate School of The Ohio State University
By
Jacob James Adams, B.S.
*****
The Ohio State University
2007
Master’s Examination Committee:
Approved by
Professor Robert Lee, Co-Adviser
Professor Ronald Xu, Co-Adviser
Co-Adviser
Co-Adviser
Graduate Program in
Electrical and Computer
Engineering
ABSTRACT
Radiofrequency ablation is an important surgical method for eliminating cancer;
however, the lack of adequate technology to image the internal organ temperature
profile forces surgeons to often guess at the ablation margin. If a sufficient temperature is not reached and all of the cancerous tissue is not destroyed, a recurrence is
likely. Therefore, we propose to develop a numerical electromagnetic and thermal
model of radiofrequency ablation that will be used in future surgical planning. The
model is based on the finite element method and couples the electromagnetic and
thermal models by considering the electric fields as the heat source. Furthermore,
the two physical phenomena are coupled through temperature-dependent material
properties.
To verify our models, we compare them to experiments conducted on excised
bovine liver. Internal temperatures are measured with thermocouples and lesion
shape and size are compared after ablation. At the same time, we attempt to predict
surface temperature during ablation in order to investigate the possibility of correlating surface temperature to internal temperatures. During the experiments, surface
temperature was measured with an infrared camera.
Over the course of three experiments, we found that internal temperatures are
predicted with good accuracy (within 2 ◦ C) when the ablation ground plane is placed
more than 8 cm away from the electrode. If the ground plane is closer, then some
ii
error is introduced into our approximate model. Also, we found that the lesion
shape and size predicted by the simulation are similar to the lesion observed after
ablation. Finally, the simulation predictions for surface temperature were mixed.
In one case, the temperature values were predicted closely but the distribution was
somewhat different. In the other case, the isothermal contours were very similar but
the simulated temperatures were as much as 25 ◦ C above what was measured.
iii
ACKNOWLEDGMENTS
This thesis would not have been possible without the help of many people, a few
who I would like to acknowledge here. I would first like to thank my advisor, Robert
Lee, who has guided me throughout this project and my years at Ohio State. I
would also like to thank my co-advisor from the Biomedical Engineering department,
Ronald Xu, whose assistance with the experimental portion of this project was invaluable. Additionally, I owe thanks to Richard Sharp who wrote the original finite
element software that my program is based upon. Finally, I must thank my friends,
family, parents, and fiancee for their support during the course of this project and for
reminding me how to balance work and life.
iv
VITA
September 8, 1982 . . . . . . . . . . . . . . . . . . . . . . . . . . Born - Columbus, OH
2005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.S. Electrical and Computer Engineering, The Ohio State University
2005-present . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graduate Fellow, The Ohio State University.
PUBLICATIONS
Research Publications
R. Sharp, J. Adams, R. Machiraju, R. Lee, R. Crane, ”Physics-Based Subsurface
Visualization of Human Tissue,” IEEE Trans. Visualization and Computer Graphics,
vol. 13, no. 3, pp. 620-629, May/June 2007.
FIELDS OF STUDY
Major Field: Electrical and Computer Engineering
v
TABLE OF CONTENTS
Page
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
iv
Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
v
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
viii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
Chapters:
1.
2.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1
Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . .
3
Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2.1
2.2
2.3
2.4
3.
Ablation Device . . . . . . . .
Data Acquisition . . . . . . . .
Experimental Procedure . . . .
Description of the Experiments
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4
5
8
11
The Finite Element Method . . . . . . . . . . . . . . . . . . . . . . . . .
14
3.1
15
15
16
17
18
20
Finite
3.1.1
3.1.2
3.1.3
3.1.4
3.1.5
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Elements Applied to Electromagnetics . . . .
Helmholtz Equation . . . . . . . . . . . . . .
Finite Element Equations . . . . . . . . . . .
Elements and Weighting Functions . . . . . .
Elemental Equations with Universal Matrices
Source Term and Boundary Conditions . . . .
vi
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3.2
Finite Elements Applied to Heat Transfer . . . . . . . . . . . . . .
3.2.1 The Finite Element Time Domain Method . . . . . . . . . .
21
22
RF Ablation Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
4.1
4.2
4.3
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25
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36
Results and Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
5.1
5.2
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5.4
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61
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
70
4.
4.4
4.5
5.
Implementation via Computer Program . . .
Representing The Coupled Problem . . . . . .
Boundary Conditions Revisited . . . . . . . .
4.3.1 Thermal Boundary Conditions . . . .
4.3.2 Electromagnetic Boundary Conditions
4.3.3 Perfectly Matched Layer (PML) . . .
Tissue Material Properties . . . . . . . . . . .
Modeling the Source and Input Power . . . .
Input Power Control . . . . . . . . .
Internal Temperature Measurements
Power Density Measurements . . . .
Surface Temperature Measurements
Conclusions and Future Work . . . .
vii
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LIST OF TABLES
Table
2.1
4.1
Page
Dimensions (in cm) of liver and location of probes with values defined
in Figure 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
Parameters used for calculating liver material properties in equation 4.3 36
viii
LIST OF FIGURES
Figure
Page
2.1
Schematic of the optical and infrared camera setup . . . . . . . . . .
7
2.2
Placement of thermocouples and electrode inside block of liver . . . .
9
2.3
Photograph of experimental setup showing liver in foam board box
with all probes in place . . . . . . . . . . . . . . . . . . . . . . . . . .
10
4.1
The leapfrog procedure for the coupled electromagnetic-thermal problem 27
4.2
A section view of the (a) full FEM model and (b) region zoomed around
the electrode with artificial electrical insulator . . . . . . . . . . . . .
28
An illustration of the cross sectional electric fields of the electrode
quasi-TEM mode. The electric field lines are shown as black arrows
and the air bounding box is shown as a dashed line. . . . . . . . . . .
32
Power absorbed by a block of tissue vs. conductivity of the tissue. A
constant current is used and the bovine liver conductivity is marked
on the plot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
Experiment 1: comparison of different methods to calculate bulk conductivity used to control the power . . . . . . . . . . . . . . . . . . .
41
Experiment 2: comparison of different methods to calculate bulk conductivity used to control the power . . . . . . . . . . . . . . . . . . .
41
Experiment 2: effect of temperature dependent material properties on
thermocouple readings . . . . . . . . . . . . . . . . . . . . . . . . . .
43
Experment 2: effect of temperature dependent material properties on
temperature distribution . . . . . . . . . . . . . . . . . . . . . . . . .
44
4.3
5.1
5.2
5.3
5.4
5.5
ix
5.6
Experiment 1: thermocouple measurements vs numerical results . . .
47
5.7
Experiment 2: thermocouple measurements vs numerical results . . .
48
5.8
Experiment 3: thermocouple measurements vs numerical results . . .
49
5.9
Experiment 1: internal temperature in ◦ C on x = 0 plane . . . . . . .
52
5.10 Experiment 2: internal temperature in ◦ C on x = 0 plane . . . . . . .
53
5.11 Experiment 3: internal temperature in ◦ C on y = 0 plane . . . . . . .
54
5.12 Experiment 1: A section (x = 0) of the liver after ablation . . . . . .
56
5.13 Experiment 2: A section (x = 0) of the liver after ablation . . . . . .
56
5.14 Experiment 3: A section of the liver (y = 0) after ablation . . . . . .
57
5.15 Experiment 1: A zoomed plot of absorbed power density in W/m3 . .
59
5.16 Experiment 2: A zoomed plot of absorbed power density in W/m3 . .
60
5.17 Experiment 1: surface temperature (◦ C) measured with IR camera . .
62
5.18 Experiment 1: difference between simulated and measured surface temperature (◦ C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
5.19 Experiment 2: surface temperature (◦ C) measured with IR camera . .
65
5.20 Experiment 2: difference between simulated and measured surface temperature (◦ C) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
67
x
CHAPTER 1
INTRODUCTION
Radiofrequency (RF) ablation is a minimally invasive procedure used to remove
cardiac arrhythmias and destroy liver, kidney, lung, bone, prostate, and breast tumors
[1]. The cancerous tissue is destroyed by electromagnetic energy, and an applicator
with a conductive tip is used to deliver electrical current to tissue. The current flows
through the tissue, agitating the ions and causing conductive heating. A complete
circuit is formed by placing a grounding pad in contact with the patient’s body,
typically on the thighs. The procedure is performed at frequencies from 450 - 1200
kHz and the generator power delivery ranges from 5 - 200 W [2].
Ablation procedures are currently performed at both microwave and radio frequencies. At microwave frequencies, energy is transferred by electromagnetic waves
that are radiated through air and tissue. However, at radiofrequencies, energy is only
transmitted to the tissue through direct conduction. This restricts the majority of the
heating due directly to electromagnetic agitation to a few millimeters immediately
around the ablation electrode [3]. The heat then propagates by thermal conduction
to surrounding tissue. Since radiofrequency (RF) ablation depends on conducting
electrical current by direct contact, any change to the conductivity of the tissue surrounding the ablation electrode can cause a significant change in the delivered energy.
1
This effect becomes critical when the tissue surrounding the electrode approaches 100
◦
C. At this temperature, the tissue begins to boil and char, causing the conductivity
to decrease rapidly and hampering electrical conduction [3]. This limits the size of
the lesions that can be created to 2 cm [2]. To increase lesion size, the cooled-tip electrode was introduced [4], [5]. The cooled-tip electrode circulates cold water through
the interior of the electrode, cooling the tissue in direct contact with the needle.
This prevents the tissue from vaporizing as quickly and allows the coagulation zone
to increase. For example, some initial studies reported 4 cm lesions created with a
cooled-tip electrode [4, 5].
The ablation procedure can be performed as either open or laparoscopic surgery.
The length of the procedure can vary but the key factor is to expose the malignant
tissue to enough heat that it is destroyed. Tissue damage is a function both time
and temperature [1, 6]. For example, a temperature of 50 ◦ C must be applied for 4
to 6 minutes to fatally damage the tissue, while temperatures above 60 ◦ C destroy
tissue almost instantaneously [2]. At the same time, information about the internal
temperature of a subject is difficult to assess, making the determination of ablation
margin an important research topic. In fact some of the mixed results in clinical
studies of RF ablation, may be attributable to the lack of technology for monitoring
tissue temperature.
Technologies such as MRI, CT, and Ultrasound are presently employed to monitor
lesion growth during surgery [2]. Another potential imaging modality is infrared
(IR) thermography which can be performed laparoscopically [7]. In order to further
study IR thermography as a potential ablation imaging modality, we propose to
develop a computer model to simulate the ablation process. In order to confirm our
2
model’s results, we have conducted a preliminary experimental study to determine
if the simulated surface and internal temperatures will compare favorably to the
experimental values.
1.1
Organization of the Thesis
The second chapter discusses a test bed featuring optical and thermal cameras
for RF ablation experiments. It also discusses a testing protocol that was used. The
third chapter develops the general finite element method that will be used to solve
the coupled electromagnetic and heat problem. The fourth chapter discuss specific
modeling issues that arise in our finite element analysis of RF ablation in liver. The
finite element model described here is strongly motivated by the test setup. Chapter
5 presents results from the simulations, discusses their implications and limitations
and concludes the paper with recommendations for future work in the area.
3
CHAPTER 2
EXPERIMENTAL SETUP
To verify our simulations and provide further insight into the physics of ablation,
a series of experiments were conceived. One of our primary goals is to investigate
the feasibility of surface temperature detection as a means of estimating lesion size.
Thus, the experiments were set up so that surface temperatures were recorded with
an infrared camera and internal temperatures could be recorded with a series of
thermocouples.
2.1
Ablation Device
The experiments were conducted using a 200 W, 480 kHz ablation generator and
single prong, 17 gauge, cooled-tip electrode with 2 cm conductive tip (Valleylab,
Boulder, CO). The device outputs RF power, RF current, tissue impedance, and
electrode tip temperature on a digital display. The primary controls used for this
project are an on/off switch, a power control dial, and a power control switch. For all
experiments the power control switch was set to manual control. The other setting,
impedance control, causes the device to shut off when the tissue impedance increases
more than 20 Ω above the base value, indicating the onset of tissue boiling and
desiccation. With the device off, the tissue is able to rehydrate for a few seconds and
4
then the device resumes normal operation. Impedance control is typically used in
clinical applications because it allows for increased lesion size [8]. However, since the
physical processes in action are the same in the two modes, this study uses the manual
control mode to provide maximum control over the application of power during the
ablation process. During experiments, we observed that the Valleylab ablation device
outputs higher current as the tissue impedance decreases. Thus, we concluded that
the device was powered by a constant voltage source.
Being a cooled-tip device, the ablation electrode requires a continuous flow of
water through it. A pump and tube are provided with the system and were used by
filling a small container with water and placing two ice packs in the bath. The tubes
were arranged to cycle water through the needle and back into the container. Since
the water will heat up as the ablation procedure progresses, the temperature in the
container was monitored with a digital thermometer. During a 41 minute experiment,
the longest conducted, the temperature in ice bath went up only 4 ◦ C.
2.2
Data Acquisition
Since a key element of this project is the ability to predict surface temperatures,
an IRRIS 160ST infrared camera was used to measure the surface temperature. Simultaneously, an optical camera was positioned to show nearly the same angle as
the IR camera. In order to achieve such a setup, an infrared mirror was placed at
45◦ to reflect light to the IR camera, which was placed in a plane perpendicular to
the optical camera and table. A schematic of this setup is shown in Fig. 2.1. The
cameras were arranged so the viewing area of the sample was approximately 5.5 cm
5
x 7.5 cm. The images were captured using National Instruments PCI-1405 image
acquisition cards and Labview software.
The data from the IR camera can be captured only as pixel brightness values.
To obtain temperature information, the IR camera was calibrated manually. To
obtain calibration data, a plastic container was placed so that it completely filled
the viewing area and was insulated on all sides with styrofoam peanuts. A digital
thermometer and three thermocouples were placed inside the container. Then water
heated to over 70 ◦ C was pumped into the container. Once the container was filled, a
thin layer of clear plastic was placed over the water’s surface. A magnetic stirrer was
used to increase the homogeneity of the water temperature. Finally, a few frames were
captured with the IR camera. Data was captured as the water cooled from 70 to 20 ◦ C
at increments of 5 ◦ C, based on the digital thermometer reading. The data collected
was processed in Matlab by creating an individual calibration scale for each pixel.
Data from any experiment could then be read in pixel-by-pixel and interpolated to
the correct temperature based on that pixel’s calibration scale. Temperatures above
69.3 ◦ C or below 21.1 ◦ C cannot be displayed because those values represent the end
points of the calibration and there is no interpolation data beyond those points.
To further evaluate the numerical model, internal temperature was measured. A
5-channel thermocouple sensor was built using T-Type wire thermocouples (Omega
Engineering, Stamford, CT) connected to a National Instruments analog to digital
converter. Since thermocouples only measure the temperature difference between
their sensing and non-sensing junctions, it is critical to keep the non-sensing junctions
at a known and constant temperature. All of the non-sensing thermocouple junctions
were located in the same box, which was open to the air in a large room where
6
Figure 2.1: Schematic of the optical and infrared camera setup
temperature did not fluctuate quickly. Temperature data from the 5 channels were
acquired simultaneously with the image acquisitions by integrating all data acquisition
functions into a single Labview interface.
Some authors have reported noise problems when using thermocouples during RF
ablation [9] due to current being conducted from the electrode through the tissue and
into the unshielded thermocouples. Still, thermocouples were used for this project
due to their low cost and small size. During our experiments we found that data
returned by the thermocouples was meaningless while the ablation device was turned
on. Some previous studies have instead used fluoroptic probes which are RF noise
immune [10,11], but they were deemed too expensive for this preliminary study. They
could be considered for future use if probes with sufficiently small diameter can be
obtained.
7
2.3
Experimental Procedure
The experiments were designed to minimize potential error sources that could
complicate the numerical comparison and simplify the discrete model where possible.
First, bovine livers were obtained from a local slaughterhouse. The livers were kept
frozen for several days and were then cut into rectangular blocks while still frozen.
A foam board box was built to hold the liver and provide good thermal insulation so
that the insulated boundary conditions in the numerical model would be accurate.
The box was open at the top to allow the cameras to capture the liver surface. A thin
plastic sheet was placed over the surface to approximate an insulated boundary. The
plastic was found to be transparent at infrared frequencies if it was kept in contact
with the sample. Once the foam board box was prepared, the defrosted liver was
placed inside and the electrode and thermocouples were inserted into the liver.
A drawing of the placement of the electrodes and thermocouples is shown in Fig.
2.2. The ablation electrode was inserted through the foam board and into the tissue
at a known location. The thermocouples were placed in the tissue parallel to the table
but perpendicular to the electrode. Since the thermocouples were only wires, they
were not stiff enough to be inserted into the tissue directly. Instead, they were placed
by first inserting a small catheter into the tissue, removing the metal needle and then
sliding the thermocouple to the end of the plastic catheter tube. The grounding pad
was placed at the end of the liver opposite the electrode with the pad perpendicular
to the electrode. A photograph of the setup is shown in Figure 2.3.
Once the sensing elements and electrode were prepared, the ablation procedure
could begin. First, the temperature in the box holding the non-sensing thermocouple
junctions was measured using a digital thermometer. This information was entered
8
Figure 2.2: Placement of thermocouples and electrode inside block of liver
into the Labview interface as the “cold-junction temperature” input. Next the Labview virtual instrument was initiated. With the data acquisition started, the cold
water pump was turned on. To avoid the tissue dropping below room temperature
from the cold water flow, the ablation device was turned on within a few seconds of
the pump being turned on. Once the ablation device was on, the following sequence
of events were carried out every minute for the duration of the test.
1. After 20 seconds, a reading of the RF input power and cold water bath temperature was recorded
2. After 30 more seconds, another reading of the RF power and bath temperature
was recorded
9
Figure 2.3: Photograph of experimental setup showing liver in foam board box with
all probes in place
10
3. During data recording, the RF power was shut off by pressing the “off” key and
the power dial was turned to zero
4. After 10 seconds, the “on” key was pressed and the power dial was quickly
turned to its prior state
The device was shut off for the last 10 seconds of each minute so that thermocouple
data could be acquired. When the RF power was on, there was far too much noise to
acquire accurate thermocouple readings. The power on/off sequence must be followed
exactly due to safety features of the ablation device. It cannot be powered on unless
the power dial is set to zero, so it must be turned to zero, powered on, and then
returned to normal power. Before each experiment, a mark was made on the dial so
that it could be quickly turned to the correct position. Raw data from the IR camera
was processed in Matlab and interpolated to meaningful temperature data based on
the calibration discussed earlier.
2.4
Description of the Experiments
The first experiment used a 5.6 x 5.6 x 4.6 cm block of liver tissue as a test piece.
The frozen liver was originally cut to be a perfect square block but we discovered
that the defrosted liver is incapable of holding any kind of form. The foam board
box served the additional purpose of shaping the formless liver into a rectangle. The
electrode was placed in the center of a face so that symmetry conditions could be
assume. The initial input power was 10 W which steadily increased as the increasing
temperature lowered the tissue impedance. By the end of the test, the input power
had more than doubled to 24 W. The first test lasted 31 minutes.
11
As the first experiment concluded and we observed the IR camera data, we thought
that the grounding pad might have been placed too near the electrode as it looked
like some residual heat from the pad was contaminating our measurements. Thus, the
second experiment was designed with a slightly larger piece of liver in all dimensions,
with the length in the z direction (along the axis of the electrode) being nearly doubled
to 10.4 cm. Again the electrode placement was designed to allow for symmetry
conditions in the model. In this case a similar trend was observed with the initial
power beginning at 20 W and ending at 33 W after 41 minutes of ablation.
The third test was designed specifically to test how tissue boiling and charring
would affect the liver. To do this, a piece of liver similar in size to the second piece
was cut and the device was turned to full power for the length of the ablation (not
including the 10 seconds per minute of power-off to collect thermocouple readings).
We expected the test to be over quickly, thinking that the boiling would onset almost
immediately, so we moved the electrode near the surface to be sure we captured some
IR data. Because of this, the third test was not symmetric in the x (vertical) direction
but still symmetric in the y direction. The power during the third experiment ranged
from 80 W initially to a maximum of 102 W after a few minutes. Not long after
reaching the maximum, boiling and charring set in and the power dropped rapidly.
By the end of the 22 minute experiment the power was only 20 W because of the
greatly increased impedance.
Dimensional data for all the tests are given in Table 2.1
12
Experiment 1 Experiment 2 Experiment 3
w
5.6
9.0
9.25
ye
0
0
0
wt
1.5
2.2
3.0
l
5.6
10.4
11.6
ze
3.8
3.0
2.9
zt
3.6
3.5
3.0
h
4.6
6.7
8.5
xe
0
0
1.25
d
.8
1.11
1.42
Table 2.1: Dimensions (in cm) of liver and location of probes with values defined in
Figure 2.2
13
CHAPTER 3
THE FINITE ELEMENT METHOD
The finite element method (FEM) is a method for solving differential equations
that has been used quite frequently in studies of RF ablation [12–19]. The basis
of the method lies in partitioning a physical region into a finite number of regions
(i.e. finite elements). The unknown solution to the function is approximated by
weighting functions over these regions. Each weighting function is associated with
a node (in the case of scalar FEM) or an edge (in the case of tangential vector
FEM) and has a known form but an unknown scaling factor. By combining the
information about the geometry of each element, the differential equation to be solved,
and the boundary conditions, a matrix equation of the form [A]{x} = {b} can be
assembled. Then the unknown scaling factors can be solved for using various matrix
equation techniques, yielding a piecewise approximation of the continuous solution.
This chapter will discuss this approach in more detail. First the FEM equations and
boundary conditions for the electromagnetic problem will be derived, followed by the
equations and boundary conditions for the heat diffusion problem.
In the case of radiofrequency ablation, two physical processes can be modeled
with differential equations governing the propagation of electromagnetic energy and
14
heat transfer. In order to predict these two phenomena, these two differential equations need to be solved. Analytical solutions to these equations only exist for very
specific situations and certainly not for the complex cases involved with living tissue.
Therefore, it is necessary to find approximate solutions to these problems.
3.1
Finite Elements Applied to Electromagnetics
The first application of finite elements will be to electromagnetic phenomena.
Since the ablation device operates at only one frequency and the time scale of electromagnetic effects is significantly less than thermal effects, we can assume time
harmonic fields and solve in the frequency domain. As mentioned before, applying
FEM to an ablation problem is not a new idea; however, to our knowledge, all studies
to date assume a quasistatic approximation for Maxwell’s equations. The quasistatic
approximation involves solving Laplace’s equation which does not take into account
any wave effects. This approach brings the possibility that important characteristics
of the wave solution could be missed, such as impedance mismatches between the
electrode and tissue. Therefore, in our studies, we will produce fullwave solutions to
Maxwell’s equations.
3.1.1
Helmholtz Equation
The Helmholtz equation describes a steady state oscillation. A vector Helmholtz
equation can be derived from Maxwell’s equations if we assume all fields are time
harmonic fields with ejωt time dependence. Beginning with Maxwell’s equations for
time harmonic fields, we have
∇ × E = −jωµH
15
(3.1)
∇ × H = jω¯E + J
(3.2)
The complex permittivity can be written as
¯ = − j
σ
ω
(3.3)
where σ is the conductivity of material. Plugging 3.3 into 3.2, we have
∇ × H = jωE + σE + J
(3.4)
Taking the curl of both sides of 3.1, substituting 3.4 into the right hand side, and
rearranging gives the following vector Helmholtz equation:
∇×
3.1.2
1
∇ × E − ω 2 E + jωσE = −jωJ
µ
(3.5)
Finite Element Equations
First, we express the Helmholtz equation of 3.5 in its weak form by taking its dot
product with the generic vector weighting function W and and integrating over Ω,
the entire domain of the problem.
Z
dΩW · ∇ ×
Z
Z
1
∇ × E + dΩ(−ω 2 + jωσ)W · E = − dΩjωW · J
µ
Ω
Ω
(3.6)
Ω
The first integrand is then expanded using a vector identity
1Z
dΩ[(∇ × W ) · (∇ × E) − ∇ · (W × (∇ × E)]
µ
(3.7)
Ω
+(−ω 2 + jωσ)
Z
dΩW · E = −jω
Ω
Z
dΩW · J
Ω
Applying the divergence theorem, equation 3.7 can be rewritten as:
Z
1Z
2
dΩ(∇ × W ) · (∇ × E) + (−ω + jωσ) dΩW · E
µ
Ω
Ω
Z
1Z
dΓ n̂ · (W × (∇ × E)) = −jω dΩW · J
−
µ
Γ
Ω
where Γ is the boundary of the domain.
16
(3.8)
3.1.3
Elements and Weighting Functions
Now, we break the domain into a finite number of smaller elements Ωe . In three
dimensions, tetrahedra are typically chosen because they conform well to irregular geometries and their geometrical properties can be expressed in closed form. In electromagnetics, the weighting functions are chosen to be vector functions associated with
edges instead of scalar functions associated with nodes. In the past, scalar weighting
functions were found to produce non-physical solutions. Tangential vector finite elements eliminate spurious solutions because they strongly enforce the divergence-free
condition on the magnetic field [20]. Each tetrahedron supports six vector weighting
functions Wie ∀ i ∈ 0, 1, ...5, which are the Whitney basis functions defined by
W0e =
(Le0 ∇Le1 − Le1 ∇Le0 )l0e
W1e =
(Le0 ∇Le2 − Le2 ∇Le0 )l1e
W2e =
(Le0 ∇Le3 − Le3 ∇Le0 )l2e
W3e =
(Le1 ∇Le2 − Le2 ∇Le1 )l3e
W4e =
(Le3 ∇Le1 − Le1 ∇Le3 )l4e
W5e =
(Le2 ∇Le3 − Le3 ∇Le2 )l5e
(3.9)
Here lie is the length of the ith edge and Lei are the linear scalar weighting functions
defined by
Lei =
1
(aei + bei x + cei y + dei z)
e
6V
(3.10)
where V e is the volume of the element and aei , bei , cei , dei are elemental constants given
by Jin [20].
17
Equation 3.8 is also true each individual element, so we write
Z
1 Z
e
e
2 e
e
dΩ (∇ × Wi ) · (∇ × E) + (−ω + jωσ ) dΩe Wie · E
µe e
e
Ω
(3.11)
Ω
Z
1 Z
e
e
−
dΓ n̂ · (Wi × (∇ × E)) = −jω dΩe Wie · J e
µe e
e
Γ
i = 0, 1, ...5
Ω
We can approximate the electric field on the element as a superposition of basis
functions. Following the Galerkin procedure, we choose the basis functions to be the
same as the weighting functions. We can then write the electric field
E=
5
X
Wje φej
(3.12)
j=0
And substituting 3.12 into 3.11, then for all internal elements we can write in
matrix form
([S] + [T ]){φ} = {b}
(3.13)
where [S] and [T ] are square matrices, {φ} is a column vector of unknowns and {b}
is a column vector of forcing terms. We can assemble these two matrices by first
creating a 6x6 matrix for each element and then adding it to the respective global
matrix. The elemental matrix entries are computed by
Sije =
1 Z
∇ × Wi · ∇ × Wj dΩ
µe e
(3.14)
Ω
Tije
2 e
e
= (−ω + jωσ )
Z
Wi · Wj dΩ
(3.15)
Ωe
3.1.4
Elemental Equations with Universal Matrices
Although analytical expressions do exist for equations 3.14 and 3.15, these matrix
entries can be expressed more efficiently in terms of a universal matrix [21, 22]. To
do this, we first eliminate the dependence on Le3 and ∇Le3 from the basis functions in
18
3.9 by using the following property of barycentric coordinates
Le0 + Le1 + Le2 + Le3 = 1
(3.16)
Using 3.16 to eliminate Le3 and ∇Le3 , we can write 3.9 as
Wi =
2
X
Wim (Le0 , Le1 , Le2 )∇Lem
(3.17)
m=0
where Wim are coefficients which are easily determined.
Equation 3.17 can then be substituted into 3.15 to arrive at the expression
[T e ] = (−ω 2 e + jωσ e )6V e
2 X
m
X
(∇Lem · ∇Len )[T mn ]
(3.18)
m=0 n=0
[T mn ] are the six universal matrices whose entries are given by
Tijmn
1
1 + δmn Z
=
2
1−L
Z 2 1−L
Z2 −L1
(Wim Wjn + Win Wjm )dL0 dL1 dL2
L2 =0 L1 =0
(3.19)
L0 =0
where δmn is the Kronecker delta function.
We can also express the [S] matrix in terms of universal matrices. First, we take
the curl of 3.17
∇ × Wi =
2
X
Cim (Le0 , Le1 , Le2 )em
(3.20)
m=0
where
e0 = ∇L1 × ∇L2
Ci0 =
(3.21)
∂Wi2 ∂Wi1
−
∂L1
∂L2
Note that by using 3.21, e1 , e2 , Ci1 , and Ci2 can be written by cyclical rotation of
the indices.
Now substituting 3.20 into 3.15, we have
[S e ] =
2 X
m
6V e X
em · en [S mn ]
µe m=0 n=0
19
(3.22)
where the entries of the universal matrices [S mn ] can be computed as
[Sijmn ]
1−L
Z2 −L1
Z 2 1−L
1
1 + δmn Z
=
2
(Cim Cjm + Cin Cjn )dL0 dL1 dL2
L2 =0 L1 =0
3.1.5
(3.23)
L0 =0
Source Term and Boundary Conditions
The source term is the right hand side of our matrix equation ([S] + [T ])φ = b. It
can be found by expanding the right hand side of equation 3.11 in the following way.
First we substitute the vector basis functions as defined in equation 3.9 to get
bi =
−jωlie
1−L
Z n
Z1
J · (Lm ∇Ln − Ln ∇Lm )dLm dLn
(3.24)
Ln =0 Lm =0
Using the definitions of the barycentric coordinates, this can be expanded as
bi =
−jωlie
Z1
1−L
Z n
[Li (bj Jx + cj Jy + dj Jz ) − Lj (bi Jx + ci Jy + di Jz )]dLm dLn (3.25)
Ln =0 Lm =0
Integrating analytically we find the source term
bi =
−jωlie
[Jx (bi − bj ) + Jy (ci − cj ) + Jz (di − dj )]
6
(3.26)
The integral over the element boundary in equation 3.11 is key in the formulation of
finite element problems and is given by
−
1 Z
dΓe n̂ · (Wie × (∇ × E))
µe e
(3.27)
Γ
These terms are shown to cancel each other on interelement faces and they remain
only if a face is on the boundary of the entire mesh. We will consider the fields
boundary of the mesh to obey either a homogeneous Dirichlet condition
n̂ × E = 0
20
on Γd
(3.28)
or a homogeneous Neumann condition
n̂ · ∇ × E = 0
on Γn
(3.29)
where Γd ∪ Γn = Γ. Both of these conditions are enforced rather easily. Because it is
a condition on the unknown itself, the Dirichlet condition can be enforced directly by
forcing that element of the φ vector to be zero. The Neumann condition can be seen
to cause the integral of equation 3.27 to be zero, so it is enforced naturally without
any further computations.
3.2
Finite Elements Applied to Heat Transfer
The application of finite elements to heat transfer is somewhat similar to the formulation described in the previous section. There are, however, several of important
differences that should be noted.
First, heat transfer is governed by a different fundamental equation. In a solid
biological media, the equation used to model transient temperature changes is the
Pennes bioheat equation [23]
ρc
∂T
= k∇2 T + hm + hb + Q
∂t
(3.30)
where ρ is the material density, c is the specific heat, k is the thermal conductivity,
hm is the metabolic rate of heat production per volume, hb is heat transfer rate due
to blood perfusion, and Q is the rate of heat production due to other sources. When
modeling RF ablation, the metabolic heat generation can be ignored because it is
insignificant compared to the heat generated by the electric current. Typical values
for metabolic heat generation are 1000 W/m3 while electrical heating is usually two
or three orders of magnitude greater [24]. Furthermore, all of the cases we will study
21
in this paper are conducted in vitro on excised bovine livers so there is no metabolic
heat generation and with no artificial perfusion induced, the blood perfusion term
will also be ignored. Then the only heat source inside the tissue is caused by ohmic
losses in the tissue which can be expressed as
Q = σ|E|2
(3.31)
Another difference in FEM for heat transfer is the basis functions. In electromagnetics, we used tangential vector basis functions in our mesh. Since temperature is
a scalar quantity, we need only use the scalar basis functions on tetrahedra given in
equation 3.10. There also is an important distinction to make between the nature of
the two problems with respect to time. Since the electromagnetic time scales are so
much shorter than the heat diffusion time scales, we have assumed that the electromagnetic fields are at a steady state, and they were solved in the frequency domain.
However, the temperature is time dependent due to the time derivative in equation
3.30, and so we must solve a transient problem. Thus our frequency domain FEM
must be extended to the time domain via the finite element time domain (FETD)
method.
3.2.1
The Finite Element Time Domain Method
Following a similar procedure to that of section 3.1, we can find a finite element
approximation of equation 3.30 expressed for each element in matrix form as [25]
[M e ]
∂ e
T + [K e ]T e = {Qe } + {q e }
∂t
(3.32)
where [M e ] is the mass matrix
Mije =
Z
ρe ce Lei Lej dΩ
Ωe
22
(3.33)
[K e ] is the stiffness matrix
Kije
=
Z
k e ∇Lei · ∇Lej dΩ
(3.34)
Ωe
{Qe } is the vector containing heat sources and given by
Qei
=
Z
Lei QdΩ
(3.35)
Ωe
and {q} is the vector containing the boundary conditions. However, for our problem,
the boundary conditions will either be Dirichlet conditions (T = T0 ) that will be
enforced directly or homogeneous Neumann conditions ( ∂∂n̂ T = 0) which will make
{q e } zero. Examining equation 3.32, we see that there is a time derivative in the
problem. In order to discretize time we will use the following form [25]
∂ a
T n+1 − T n
T =
∂t
∆t
(3.36)
T a = (1 − θ)T n + θT n+1
s where θ is a free parameter with 0 ≤ θ ≤ 1 and T n represents the temperature at
the nth time step. If we choose θ = 0, we have the backward difference method that
is explicit (i.e. the finite element matrix does not need to be inverted). However,
this method only exhibits O(∆t) accuracy and is stable only when ∆t is chosen to
be smaller than a maximum time step. A more accurate and unconditionally stable
approximation is the Crank-Nicholson method where θ = 0.5. This method exhibits
O(∆t2 ) accuracy [25] and is used in this study.
Choosing the appropriate time step is critical to ensure an accurate and stable
solution. Since the Crank-Nicholson method is unconditionally stable, the time step,
no matter how large, will never cause the solution to experience non-physical, unbounded growth. However, the accuracy of the solution is directly related to the size
23
of the time step used. To test the convergence of our model, the same simulation was
run at several time steps. Due to the experimental procedure described in Chapter
2, the maximum time step that could be used was 10 seconds. Simulations were run
with time steps of 1, 5, and 10 seconds. The temperatures computed when using each
time step were compared at three different points. After a 41 minute simulation, the
temperatures measured were within .03◦ C, and the temperature history curves were
indistinguishable from one another. The standard time step used for our simulations
was chosen to be 5 seconds. Although 10 seconds yielded the same result with fewer
computations, a 5 second step allowed us more flexibility in defining power on/off
events if needed.
Finally, it should be noted that a minimum time step also exists for parabolic
equations like the heat diffusion equation [26, 27]. Below the minimum time step
damped oscillations occur. Therefore, while the answer does eventually converge
to the correct one, it initially yields non-physical temperature values. This is a
potential error source in our model since material properties are calculated based
on temperature. A large oscillation could result in incorrect material properties, and
these errors would then be propagated through time even after the oscillations have
died out. Our model was also tested for these oscillations. Small oscillations did occur,
even at our maximum step size of 10 seconds. However, we found that the boundary
conditions and internal sources did not encourage the rise of these oscillations. The
oscillations tend to occur more strongly when there is a large jump in temperature
or internal power density between adjacent nodes.
24
CHAPTER 4
RF ABLATION MODELING
Today, the finite element method is a well established method and deriving equations for the elemental matrices is a rote process. However, finding the equations
is only the first step in developing a working model. All inputs to the algorithm
must be carefully designed to produce a model that accurately reflects the physics of
the problem. The design of our model is motivated by the experiments described in
Chapter 2.
4.1
Implementation via Computer Program
Although other studies have used commercial software such as Ansys [28], ETherm
[29], and Comsol [13] to solve similar finite element problems, we opted to create
custom software because it allows for greater flexibility both in problem definition
and data processing. Thus, a large portion of the work for this project was spent
writing a program to build a database representing the finite element mesh, solve
the problem, and display and record data. The program was written in C++ on the
Cygwin platform and a user interface was created with the Fast Light Toolkit (FLTK).
The finite element method produces a sparse matrix which was stored in compressed
sparse form to reduce memory usage and make the method feasible. The matrix
25
system was solved directly after LU factorization with the asymmetric, complex linear
algebra package UMFPACK [30]. While our software was capable of visualizing the
raw data, most of the plots were created in Matlab using data exported from the
custom program. The other important step that cannot be performed with the custom
software is the creation and meshing of geometry. Instead, the model was created in
Ansys and a tetrahedral mesh was generated and exported to a format that could be
used as an input to our software.
4.2
Representing The Coupled Problem
Since the heat source in the thermal model is due to the electromagnetic fields,
the two solutions are closely coupled. Moreover, both the electrical and thermal
properties of the tissue are temperature dependent. One approach to representing
this coupling in a discrete solution is a leap-frogging approach in which the electromagnetic and thermal solutions are solved in alternating succession. To effectively
model the feedback loop between the heat and electromagnetic solvers, the leapfrogging algorithm shown in Figure 4.1 was implemented. The procedure consists of
compute the electromagnetic fields and using them to create a source for the heat
transfer problem. Then, the heat problem is solved for as much as 30 seconds of
simulation time. After heat transfer, the temperature in each element is observed,
and the temperature-dependent material properties are adjusted according to several
models which will be discussed later in this chapter. As previously mentioned, the
time scale required to reach a steady state solution in the electromagnetic problem is
much smaller than the time required for an accurate thermal solution. Therefore, the
26
Figure 4.1: The leapfrog procedure for the coupled electromagnetic-thermal problem
electromagnetic fields can be solved for as a steady state solution in the frequency
domain while the temperature must be solved in the time domain.
4.3
Boundary Conditions Revisited
The boundary conditions are one critical input that will determine both the accuracy and complexity of the model. A discrete model must be bounded, and for
every boundary in the problem, a condition must be specified and implemented in
the code. Some of the simplest conditions are described in Chapter 3, but more
complex conditions exist. Additionally, conditions are needed for both the electrical
and thermal models and each model will have independent boundary conditions. The
first two experiments were symmetric about the axis of the electrode. This is advantageous because only one quarter of the geometry needs to be simulated, greatly
reducing memory requirements and computational time. A section of the computational model is shown in Figure 4.2(a). The figure shows the three main materials
in the model: the tissue, an air bounding box, and a perfectly matched layer (PML)
which will be discussed in the following section. The figure also shows the symmetry
plane; only the part of the model to the right of the symmetry plane was created and
meshed.
27
(a)
(b)
Figure 4.2: A section view of the (a) full FEM model and (b) region zoomed around
the electrode with artificial electrical insulator
28
4.3.1
Thermal Boundary Conditions
For this study, the temperature distribution was only important inside the tissue.
Thus, the thermal model was truncated at the edge of the tissue. The electrical insulator shown in Figure 4.2(b) was the only material other than the tissue in the domain
of the thermal model. The block of tissue then has several boundaries: symmetry
planes, outer walls, a top and bottom wall, and a curved boundary where the electrode touches the tissue. To represent the symmetry planes, homogeneous Neumann
boundary conditions were used. In the experiments, the outer walls were carefully insulated using thick foam board, so an insulating boundary condition closely represents
the physical situation.
The bottom wall was also covered by foam board, but the grounding pad was
placed between the tissue and the foam. Since all of the electric current passes into
the grounding pad, it may cause some boundary heat flux. The top wall was open so
that the IR camera could capture surface temperature, although it was covered with
a sheet of clear plastic, providing some insulation. Some convective heat transport is
likely to occur at this boundary, but we assumed it was negligible. Despite the slightly
more complicated situations at these last two boundaries, the top and bottom walls
were also approximated as insulating boundaries. The insulating boundary condition
used for all of these walls is the zero normal heat flux condition
∂
T
∂ n̂
= 0, which
is observed to be the same as the homogeneous Neumann condition used on the
symmetry planes.
Finally, the boundary at the electrode-tissue interface was dealt with by setting a
Dirichlet (constant temperature) condition. This interface cannot support insulating
conditions because the cold water flow through the needle results in heat being carried
29
away from the tissue. During the experiments it was observed that the electrode tip
temperature readings did not diverge much from 19 ◦ C, so a constant temperature
approximation was set on the interface. A similar approach has been carried out in
other studies of cooled-tip electrodes [17, 31, 32].
4.3.2
Electromagnetic Boundary Conditions
In order to properly model the electromagnetic fields, the air and PML regions
need to be included in the model in addition to the tissue. The PML is an artificial
material that will absorbing incoming electromagnetic waves. It will be discussed
later in this section.
Because we need to consider a larger area, the boundaries for this model are
slightly different than the thermal model. There are still two symmetry planes that
must be represented by boundary conditions. The electrical current source in this
problem is parallel to the symmetry planes. Using electromagnetic image theory [33],
we can reflect this source and the rest of the geometry about the planes by applying perfect magnetic conductor (PMC) conditions. On a PMC surface a Dirichlet
condition on the magnetic field exists, n̂ × H = 0, which is equivalently a Neumann
boundary condition on the electric field. Another surface that must be considered is
the bottom surface that contacts the ground plane. Due to its excellent conductive
properties, this surface can be approximated as a perfect electric conductor (PEC),
leading to a Dirichlet condition of the form n̂ × E = 0 which can be enforced directly.
Similarly, the electrode can also be approximated as a perfect conductor and a PEC
condition can be set on the electrode-tissue boundary. The boundary condition at
the top of the model (on the far side of the PML) is not of particular importance
30
because the PML should attenuate the electric fields so that no field is present at
the top boundary. The final boundaries to consider are the boundaries on the air
bounding box. These are artificial boundaries because they are open in the physical
setup. These boundaries could be dealt with by placing a PML along the sides of
the air box. However, this greatly increases the number of unknowns in the problem.
Instead, note that the electrode is a poor radiator at radiofrequencies because it is
much smaller than a wavelength (over 600 meters). Hence, very little of the electric
field is traveling away from the electrode towards these boundaries through the air;
most of the fields along the electrode, into the tissue and then the grounding pad.
It is reasonable then to assume that an approximate boundary condition could be
substituted for the PML. Two simple types of conditions are possible, the PMC condition or the PEC condition. The unperturbed mode formed by the electrode in free
space is a quasi-transverse electromagnetic (TEM) mode illustrated in Figure 4.3.
The PMC boundary condition causes all electric fields to be parallel to the boundary
while the PEC boundary condition causes all electric fields to be normal to the boundary. Clearly, from looking at Figure 4.3, it can be seen that most of the electric fields
are normal to the air bounding box. Therefore, it is expected that a PEC boundary
condition would cause the smallest perturbation to the actual solution. Both PEC
and PMC boundary conditions were compared to the solution with PML on the sides
and as expected, the energy deposition in the tissue matched most closely when the
PEC condition was applied.
31
Figure 4.3: An illustration of the cross sectional electric fields of the electrode quasiTEM mode. The electric field lines are shown as black arrows and the air bounding
box is shown as a dashed line.
4.3.3
Perfectly Matched Layer (PML)
When considering open domain electromagnetic problems solved with a finite
method like the finite element method, the mesh must be truncated. In order to accurately simulate an open domain, the boundary must absorb electromagnetic waves
without reflecting them. One way to create this effect is with a perfectly matched
layer (PML) [34]. The PML is an artificial material that is added beyond the truncated boundary. The layer is perfectly matched in the sense that a wave traveling
in free space experiences no reflection at the air-PML boundary. At the same time,
the wave is rapidly attenuated inside the PML so that by the time it reaches the end
of the PML, it is negligible so the reflection at the end of the PML is insignificant.
This matching and attenuating effect is achieved with a non-physical lossy anisotropic
medium. Both the permittivity and permeability of the medium are anisotropic, and
32
they are each carefully designed to create the matching and attenuating effect. For
example, the PML properties required to absorb a wave traveling in the ẑ direction
are represented as
= 0 [Λ]
(4.1)
µ = µ0 [Λ]
where
a − ja
0

0
a − ja
[Λ] = 
0
0

0
0
1
a−ja



(4.2)
As shown in Figure 4.2(a) the PML is included at the point where the electrode leaves
the air box. The wave traveling along the electrode continues into the PML and is
eventually dissipated.
In choosing the attenuation properties of the PML, two things must be considered.
A smaller attenuation constant may not attenuate the incident waves sufficiently,
allowing them to reflect off the back of the PML and return to the physical domain.
At the same time, too large an attenuation constant will cause numerical reflections
at the air-PML interface because the rapid attenuation cannot be accurately modeled
without a smaller mesh size. In our case the PML region is 10 cm long (approximately
λ/6000) and the PML mesh size is on the order of λ/10000. Few parametric studies
have been conducted at these electrically small distances, so our PML was tested for
the optimal attenuation value. To do this, the PML attenuation value was adjusted
upwards from small values until the exponential decay of the fields inside the PML
was observed to reduce incident fields to negligible values at the far side of the PML.
Also, the value was kept small enough that most of the field was initially transmitted
into the PML, rather than being reflected due to insufficient mesh size. Based on
33
these tests, the value chosen for our experiments was a = 5000 with a is defined in
equation 4.2.
4.4
Tissue Material Properties
Another important consideration in this work is the properties of tissue and dependence on temperature. Several properties are needed for the electromagnetic and
thermal simulations including relative permittivity r , electrical conductivity σ, thermal conductivity k, specific heat c, and density ρ. Previous studies of RF ablation
have often assumed generic temperature dependence of electrical conductivity, typically +2%/◦ C [17] while others have included more complex models [35]. The other
parameters are typically held constant, although some studies have introduced a temperature dependence for thermal conductivity during cardiac ablation [28].
The value of electrical conductivity is key because it determines the power drawn
from the device. If σ increases, the impedance of the tissue will decrease and the
constant-voltage ablation source will deliver more power. Pop et al recently studied
the electrical properties of ex vivo porcine kidney [36]. They investigated the temporary and permanent effects of heating on the electromagnetic properties of kidney
cortex at 460 kHz, applying an Arrhenius model to both the conductivity and permittivity. At present, this is closest model available for bovine liver at our desired
frequency (480 kHz). While kidney and liver do differ slightly in electrical properties,
they are within an order of magnitude; both the conductivity and permittivity of
liver tend to be about half of that of kidney at radiofrequencies [37]. It has also been
shown that bovine and porcine liver and kidney display similar properties [38, 39].
34
Therefore, we will use the model developed for the porcine kidney as a model for the
properties of the bovine liver.
The Arrhenius model used suggests that damage accumulates as the tissue is
exposed to heat over time and eventually causes a permanent change in the material
properties. This is a similar model to those used in studying the extent of tissue
injury [1,6]. The paper modeled kidney r and σ as having both a linear temperature
dependent component and a thermal damage component with the following equation
−C3
x(t, T ) = x0 [1 + C1 (T (t) − T0 ) + C2 (1 − e
Rt
0
−Ea
e RT (τ ) dτ
)]
(4.3)
where T0 = 22◦ C and R is the gas constant (1.98 cal mol−1 K−1 ). C1 , C2 , C3 are
constants and Ea is the activation energy. The values used for these constants in
our liver model are shown in Table 4.1. For our model, x0 is half of the amount
given in [36] because the kidney tends to have higher conductivity and permittivity
than liver. Note that the C2 is negative for r and positive for σ. This means the
conductivity is increases with increasing temperature while the permittivity tends to
decrease.
The thermal conductivity, k, is also considered temperature dependent. Although
an Arrhenius model similar to that of [36] likely holds true for k, no equivalent study
was found in the literature. Instead, a linear model developed by Bhattacharya and
Mahajan [40] is used in this study
k(T ) = .4475 + .0033T
with k(T ) in W/m-K.
35
(4.4)
x
r
σ
x0
C1
C2
C3
1605 .013 −.57 5.85 × 1028
.11 .016
.63 5.73 × 1034
Ea
48.32 × 103
57.42 × 103
Table 4.1: Parameters used for calculating liver material properties in equation 4.3
Bhattacharya and Mahajan noted that irreversible changes to thermal conductivity did not occur until around 80 ◦ C. This is in contrast to the results of [36] where
permanent changes in electrical properties were seen as low as 50 ◦ C.
In our model, the specific heat and density are considered constant, and the thermal diffusivity α =
k
ρc
then has the same linear temperature dependence as the
thermal conductivity. However, it has been shown that in many biomaterials, the ρc
term is not constant with temperature, and the thermal diffusivity increases an order
of magnitude more quickly with temperature than the conductivity alone [41]. This
suggests that the product of density and specific heat is decreasing with temperature,
rather than constant as in our model. Valvano measured both diffusivity and conductivity up to 45 ◦ C, so his results are limited by the low upper temperature bound
while Bhattacharya and Mahajan’s model is valid up to 80 ◦ C. A combination of
the two models could be used in the future to more accurately model the changes in
thermal properties.
4.5
Modeling the Source and Input Power
Since the Valleylab ablation device is a constant voltage rather than constant
power device, the power changes as the tissue impedance changes and consequently,
36
as the temperature profile changes. Proper modeling of the source is critical so
that the input power will be correct after each time step. In order to model the
source, information is needed about the source impedance of the generator and feed
mechanism. We contacted Valleylab but could not obtain this information. Without
knowing the details of the source, we tried to model the input power as closely as
we could. To approximate the current flowing down the electrode, we created a gap
current source by replacing a small section of the electrode with air and defining a J
source in the gap directed along the electrode axis.
37
CHAPTER 5
RESULTS AND CONCLUSION
Three experiments were conducted using a similar setup (see Chapter 2). Experiment 1 and Experiment 2 were similar except that the distance between the ground
pad and liver was increased in the second experiment. In both experiments, the
geometry was symmetric about in xz and yz planes through the electrode axis. In
Experiment 3, we wished to test if the simulations would fail when the boiling and
charring that begins for temperatures over about 80 ◦ C occurred. In order to achieve
these high temperatures, the ablation device was turned to full power for Experiment
3. In this experiment, we also disrupted the symmetry of the model by move the
electrode slightly closer to the surface being measured by the infrared camera. Experiment 1 was run for 30 minutes, Experiment 2 for 41 minutes and Experiment 3
for 22 minutes.
For comparison, three FE models were created that approximated the slightly nonuniform shape of the livers as perfect rectangles. The method and specific parameters
used in these simulations are detailed in Chapters 3 and 4.
The goal of this study was to determine if our model can simulate the electromagnetic and thermal processes that govern RF ablation. First, we investigated if we
could develop a model that could predict the power input into tissue based only on
38
geometry. Next, we investigated the internal temperature of the models to see if they
would correlate with the thermocouple readings. Finally, we compared the surface
temperatures of our model to the surface temperatures measured with the infrared
camera.
5.1
Input Power Control
While the actual device is fed by a voltage source inside the case, the input power
was modeled with a gap current feed on the electrode. Unfortunately, this simple
approximation to the actual source did not behave like the device. Instead, only the
impedance mismatch between the electrode and tissues was important. In numerical
experiments where the conductivity of the tissue was varied, the power absorbed
reached a maximum when the tissue had a conductivity of about .05 S/m. Then
if the conductivity was increased from that point, the power would drop. A plot of
the power absorbed in this numerical test is shown in Figure 5.1. In contrast, while
using the actual device we always observed the power to increase when the impedance
decreased (i.e. conductivity increased).
To rectify this difference, we attempted to couple the input current to the conductivity of the tissue. Then as the conductivity increased, by simultaneously decreasing
the absorbed power we could boost the input current by the proper amount to simulate the increase in power that actually occurs. Several methods of determining an
appropriate current scaling factor or ”bulk” conductivity were tried. Most were based
on the idea that the conductivity near the tip of the electrode was the most critical to
determining the bulk impedance that the ablation device sees. The first method takes
an average of the conductivity of all the elements weighted by their distance from the
39
Figure 5.1: Power absorbed by a block of tissue vs. conductivity of the tissue. A
constant current is used and the bovine liver conductivity is marked on the plot.
tip of the electrode. Also attempted were weighting by this distance squared. Next,
a similar method was employed except that the conductivities were weighted by the
strength of the electric field in each element rather than their distance. Finally, a box
was chosen around the electrode and the conductivities of all elements within the box
were averaged. The result of these methods applied to the simulation of Experiment
1 and Experiment 2 are shown in Figure 5.2 and 5.3, respectively.
It is clear that none of these methods worked consistently. While the r2 method
and box method worked fairly well for Experiment 2, they both failed badly for
Experiment 1. Therefore, until a better feed model for the device can be developed,
we decided to force the power to follow the power recorded in the experiments. This
was done by calculating the electric fields with an initial feed current and the power
absorbed using that current. Then the difference between the simulated power and
40
Figure 5.2: Experiment 1: comparison of different methods to calculate bulk conductivity used to control the power
Figure 5.3: Experiment 2: comparison of different methods to calculate bulk conductivity used to control the power
41
the actual power was computed and the feed current was re-scaled. The electric fields
and absorbed power were computed again and the current re-scaled until the correct
power was achieved. This process was carried out at every power data point (30
seconds of simulation time). The remainder of the data was generated by using this
forced-power approach.
With this forced-power method, the effects of the temperature-dependent model
were investigated. Using the geometry in Experiment 2, we disabled the temperature
dependence of the tissue properties and compared it to the normal, tissue dependent
model. Temperature history was taken at three points and the results are shown in
Figure 5.4. Additionally, the temperature profiles after 41 minutes are compared in
Figure 5.5. In the temperature history plot there is almost no difference at all when
temperature dependence is disabled. The temperature profile shows some small differences. Without temperature dependence, the heat seems to transfer out in a more
isotropic way. When temperature dependence is enabled, the heat tends to transfer
quickest in the direction that is heated initially because the thermal conductivity is
increased. This leads to an anisotropic pattern like that in Figure 5.5(b) where more
heat is transferring away from the electrode, along its axis. The lesion dimension is
smaller with temperature dependence because the heat is being transfered away from
the “lesion area” more quickly and spreading deeper into the tissue. So temperature
dependent properties do not play a large role when the power is forced to a certain
value, however, once an adequate model of the source is developed, the temperature
dependence of tissue will become critical since the material properties will control the
input power.
42
Figure 5.4: Experiment 2: effect of temperature dependent material properties on
thermocouple readings
5.2
Internal Temperature Measurements
The comparison of internal thermocouple measurements to simulated results also
provides some indication of the accuracy of the model. The results for Experiment
1 are shown in Figure 5.6. Thermocouples 1 and 5 were placed symmetrically about
the yz plane as were thermocouples 2 and 4 (see Figure 2.2). Thus, each pair is
shown on the same plot. The measured results differ significantly from the simulated
results in two out of the three locations. Also, there is no consistent pattern in the
deviation from the measured value. For thermocouples 1 and 5 the simulated value
is 15 ◦ C too high and for thermocouple 3, it is 10 ◦ C too low. The mixed results for
this experiment may be attributable to the incorrect modeling of the ground plane.
Since it is not electrically connected to the far end of the electrode to form a complete
43
(a) Constant material properties
(b) Temperature-dependent material properties
Figure 5.5: Experment 2: effect of temperature dependent material properties on
temperature distribution
44
circuit, we are not actually modeling a grounded object, but a freestanding slab of
metal. In terms of electromagnetics, this is a completely different situation than a
ground that is connected to the the negative terminal of a source. Therefore, since
the electrode is very close (less than 2 cm) to the “ground plane”, the approximation
greatly affects the solution. In fact, for the cases where the ground plane is on the
order of 8 cm away from the electrode, the internal temperature results are quite
accurate.
A comparison of the thermocouple measurements during Experiment 2, shown
in Figure 5.7, looks significantly better. Again, thermocouples 1 and 5 are plotted
together as are thermocouples 2 and 4. The experiment lasted 41 minutes and by
the end of the test, the average error between the thermocouples and simulated data
was only 2.4 ◦ C. At times under 15 minutes, the temperature differences are quite
small, less than 1 ◦ C. The measured data from Thermocouple 4 were extremely
noisy, although a general trend can still be seen from the plot. The variance is
suspected to have been caused by a loose connection or bad junction since it a similar
problem occurred during the first and third experiments. These measurements may
be more accurate than the first experiment because the ground plane is further away,
lessening the effect of the inaccurate ground plane model. Note that the temperature
simulations in Experiment 2 are all slightly higher than the measured readings. Power
is being lost somewhere which is not being modeled properly. One suggestion is that
the insulated boundary conditions are inaccurate on the open surface and heat should
be lost on those boundaries. Alternatively, not all of the RF power that the machine
outputs is making it into the tissue. A small percentage of it could be radiated into
space, causing slightly less heating to the tissue than expected.
45
In Experiment 3, the thermocouples could not be considered symmetrical because
the electrode was shifted. Therefore, all 5 thermocouples are compared individually
to the simulated results in Figure 5.8. Thermocouples 1, 3, and 4 display excellent
agreement with the measured results. The differences in final temperature are very
small, with thermocouple 4 having the highest error of about 2◦ C. Thermocouple
2 differs by a large amount, but the differences can be attributed to experimental
error. Thermocouple 2 is the closest to the electrode so it should record the highest
temperature. In the simulated results, this holds true but in the experimental results,
both thermocouple 1 and 3 recorded higher temperatures. This suggests that during
the experiment, thermocouple 2 made poor contact with the tissue, resulting in a lower
temperature reading. The data collected by thermocouple 5 are rendered meaningless
because they are extremely noisy. The cause is probably a poor connection, either in
the thermocouple junction box or at the temperature-sensing junction itself.
The simulation provides the additional benefit of being able to view internal temperature and power absorption data that cannot be measured during the experiment.
Figure 5.9 and Figure 5.10 show the propagation of heat over time throughout the
liver during Experiments 1 and 2. The slices are taken at the x = 0 plane so the
electrode is also sectioned and appears in the lower left corner. No temperature data
exists inside the electrode so it is displayed as zero. Isothermal contours are shown
every 5 ◦ C. The temperature profile in Experiment 1 differs significantly from that
in Experiment 2. It again appears that the proximity of the ground pad significantly
distorts the temperature profile by drawing most of the current towards the pad.
While the pattern in Experiment 2 looks like the typical prolate spheroid we would
expect, the pattern of Experiment 1 looks some what like a very wide spheroid that
46
(a) Thermocouples 1 and 5
(b) Thermocouples 2 and 4
(c) Thermocouple 3
Figure 5.6: Experiment 1: thermocouple measurements vs numerical results
47
(a) Thermocouple 1 and 5
(b) Thermocouple 2 and 4
(c) Thermocouple 3
Figure 5.7: Experiment 2: thermocouple measurements vs numerical results
48
(a) Thermocouple 1
(b) Thermocouple 2
(c) Thermocouple 3
Continued
Figure 5.8: Experiment 3: thermocouple measurements vs numerical results
49
Figure 5.8 (continued)
(a) Thermocouple 4
(b) Thermocouple 5
50
has been cut in half. Experiment 1 also shows a maximum temperature immediately
below the needle tip that is 20 ◦ C higher than Experiment 2.
The internal temperature contours of Experiment 3 are shown in Figure 5.11.
These sections are taken along the y = 0 plane. The temperature maps look similar
to the prolate spheroid seen in Experiment 2. This is expected because the ground
plane is sufficiently far away from electrode in both cases, and any errors introduced
by the ground plane model are negligible. The distinguishing feature of the third
experiment is that the temperature around the needle rises above 70 ◦ C within 1
minute due to the very high power being used (80 - 100 W). In this temperature
range, our models for the material properties are no longer valid. However, it is
interesting to note that the thermocouple readings still aligned well with the simulation results despite the incorrect material properties. This is further evidence that
the temperature dependent material properties are not significant when not used to
determine the input power. Observing the temperature profile at 15 and 22 minutes,
we see that the 70 ◦ C contour has contracted. By the time 15 minutes have passed,
the tissue is mostly desiccated and the impedance is quite high. Because of this, the
power delivered peaks in the first few minutes and then declines quickly so that by
the end the device was only delivering 20 W. Since the input power is so low, the
temperatures near the electrode fall while the heat continues to spread slowly into
the tissue.
As a first order approximation, we can assume that the coagulation necrosis region
is anywhere within the 50 ◦ C contour. In this case, after 30 minutes, the simulation
predicts that the lesion should be the entire length of the liver in the y direction.
Figure 5.12 shows a section taken after Experiment 1, along the same plane shown in
51
(a) 1 min
(b) 5 min
(c) 10 min
(d) 20 min
(e) 30 min
Figure 5.9: Experiment 1: internal temperature in ◦ C on x = 0 plane
52
(a) 1 min
(b) 5 min
(c) 10 min
(d) 20 min
(e) 30 min
(f) 41 min
Figure 5.10: Experiment 2: internal temperature in ◦ C on x = 0 plane
53
(a) 1 min
(b) 5 min
(c) 10 min
(d) 15 min
(e) 22 min
Figure 5.11: Experiment 3: internal temperature in ◦ C on y = 0 plane
54
Figure 5.9. The photograph displays a similar coagulation pattern to that predicted
by the simulation. Most of the necrosis is concentrated in an arc that goes from the
electrode to the ground pad and nearly the entire liver is coagulated in the y (left &
right) direction.
In Figure 5.10, the simulated lesion is a prolate spheroid with dimensions 3.6
x 3.1 cm in the axial and radial directions, respectively. This can be compared to
the section taken after Experiment 2, shown in Figure 5.13. The dimensions of the
coagulation region are estimated to be 3.5 cm x 2.8 cm, which is similar to the model.
Finally, a section of the liver after Experiment 3 is shown in Figure 5.14. The
coagulation region is estimated to be 4 cm x 3 cm in the photograph. From Figure
5.11, we estimate the lesion size to be 4.5 cm x 3.5 cm. Again the experimental results
are similar to the simulations, although the size differs the most in Experiment 3. The
rapid heating caused by the high power invalidates our material model, and one effect
may be that the lesion size is over-estimated. Heat may be spreading too quickly due
to the very high thermal conductivity resulting from the linear model.
5.3
Power Density Measurements
Another quantity that can be measured with the simulations is the power density
in the tissue. Figures 5.15 and 5.16 show the power density in the liver at the x =
0 plane with the electrode protruding up from the lower left corner. Note that the
plots are zoomed in significantly compared to the temperature plots but they are both
on the same scales for cross comparison. Clearly, the power distribution inside the
liver looks quite different from the temperature distribution. However, it is clear why
the temperature distributions look so different by observing the power distributions.
55
Figure 5.12: Experiment 1: A section (x = 0) of the liver after ablation
Figure 5.13: Experiment 2: A section (x = 0) of the liver after ablation
56
Figure 5.14: Experiment 3: A section of the liver (y = 0) after ablation
In Experiment 1, almost all of the power is concentrated just beyond the tip of the
electrode. The heat diffuses out evenly from there, creating a pattern that is strongly
shifted towards the grounding pad. On the other hand, in Experiment 2, there are
two major source points of conductive heating. One is just to the side of the electrode
tip and a smaller source is just to the side of the other end of the conductive tip.
This heating pattern tends to create a more circular temperature distribution.
It is interesting to note that even though the electric fields only exist in a small
region within about 5 mm of the electrode, the temperature is able to diffuse much
further away. The overall power increase over the experiment period (from 20 to 33
W) is also evident in the figure as the high power density area grows wider. However,
despite changing material properties, the distribution of power does not change very
57
much. The patterns do not spread out much with time but rather the peaks get
higher.
5.4
Surface Temperature Measurements
Another goal of this project is to investigate if the surface temperature can be
predicted using a numerical model and whether that temperature gives any information about the internal temperatures. The plots in Figures 5.17 and 5.19 compare the
simulated surface temperature (left) and the measured surface temperature (right) at
several time points.
For Experiment 1, the viewing area of the IR camera is the entire surface of the
liver, 5.6 x 5.6 cm. At the early times, when the surface temperature is little different
than the ambient temperature, the model predicts quite closely both the temperature
and the shape of the distribution. However, once the surface temperature becomes
significantly different than the ambient temperature, such as in Figures 5.17(e),(g),
then the simulated temperature significantly overestimates the temperature,although
the shape of the distribution is similar. In some places it overestimates the surface
temperature by as much as 25 ◦ C as seen in Figure 5.18. Again, it is likely that this
error is caused by a poor approximation of the boundary as an insulating boundary
which loses no heat. If it were instead modeled as a convective boundary, then the
boundary would lose an amount of heat proportional to the temperature difference
between the surface temperature and ambient temperature.
For Experiment 2, the viewing area of the IR camera was approximately 7 cm x
7.6 cm and since the symmetric structure was only simulated on one side, a 3.5 cm
x 7.6 cm viewing window is shown. Again, isothermal contours appear every 5 ◦ C
58
(a) 1 min
(b) 5 min
(c) 10 min
(d) 20 min
(e) 30 min
Figure 5.15: Experiment 1: A zoomed plot of absorbed power density in W/m3
59
(a) 1 min
(b) 5 min
(c) 10 min
(d) 20 min
(e) 30 min
(f) 41 min
Figure 5.16: Experiment 2: A zoomed plot of absorbed power density in W/m3
60
and the electrode protrudes up from the lower left corner. The first three time points
(a-f) do not yield much information. The surface temperature is very close to room
temperature and it is difficult to tell whether the heating on the upper right corner
of the infrared is noise or a genuine temperature increase. Since the spot seems to
increase steadily with time, it is probably a real increase. It is unclear why heating
would be occurring far away from the electrode first, but it could be caused by natural
inhomogeneity in the liver such as a large blood vessel. The last three time points
provide more insight into the behavior of the pattern. In the simulated results, the
peak surface temperature is centered immediately above the tip of the needle and the
heat propagates out from there in a fairly circular pattern. The temperature pattern
in the measured data shows a similar range of temperatures though some features
of the pattern differ. The experimentally measured temperature spreads away from
the electrode more quickly than the simulated temperature does, so the contours
appear different. While the simulated contours appear circular, the contours in IR
images appear parabolic or ovular. Figure 5.20 shows the amount the simulated data
deviated from the measured data. The maximum error was only 5 ◦ C. The simulation
predicted higher temperatures in the area above the needle while predicting lower
temperatures in the area farthest from the needle.
5.5
Conclusions and Future Work
We have built a custom finite element model for RF ablation featuring a fullwave
electromagnetic solver coupled to a thermal time-domain solver. We implemented a
leap-frogging, transient algorithm to couple the electromagnetic source to the load.
61
(a) 5 min, Simulation
(b) 5 min, Experiment
(c) 10 min, Simulation
(d) 10 min, Experiment
Continued
Figure 5.17: Experiment 1: surface temperature (◦ C) measured with IR camera
62
Figure 5.17 (continued)
(a) 20 min, Simulation
(b) 20 min, Experiment
(c) 30 min, Simulation
(d) 30 min, Experiment
63
(a) 5 min
(b) 10 min
(c) 20 min
(d) 30 min
Figure 5.18: Experiment 1: difference between simulated and measured surface temperature (◦ C)
64
(a) 1 min, Simulation
(b) 1 min, Experiment
(c) 5 min, Simulation
(d) 5 min, Experiment
(e) 10 min, Simulation
(f) 10 min, Experiment
Continued
Figure 5.19: Experiment 2: surface temperature (◦ C) measured with IR camera
65
Figure 5.19 (continued)
(a) 20 min, Simulation
(b) 20 min, Experiment
(c) 30 min, Simulation
(d) 30 min, Experiment
(e) 41 min, Simulation
(f) 41 min, Experiment
66
(a) 1 min
(b) 5 min
(c) 10 min
(d) 20 min
(e) 30 min
(f) 41 min
Figure 5.20: Experiment 2: difference between simulated and measured surface temperature (◦ C)
67
Furthermore, we included a temperature-dependent model for the electrical and thermal properties of bovine liver.
The purpose of the model is ultimately to investigate new imaging modalities for
radiofrequency ablation, such as infrared thermography. The first step in achieving a
working model is to verify the model with experimental data. Thus, we constructed
a benchtop setup up for in vitro RF ablation of biological tissue. The setup consisted
of an optical camera for visualization and positioning, an infrared camera for imaging
temperature on the tissue surface, and a 5 channel thermocouple sensor for measuring
internal temperature. We conducted three ablation tests, and created three distinct
finite element models to go along with each test. Results from the simulations and
experiments were correlated and compared. The comparison yielded mixed results.
While the thermocouple data for two tests came within a 2 ◦ C margin, in another
test the simulation was as much as 15 ◦ C off. Qualitatively, the internal temperature profiles appeared to align well with the coagulation regions observed when the
ablation tissue was sectioned. The surface temperature comparison looked promising
although inaccurate thermal boundary conditions appear to be keeping the results
from being even better. Good agreement in surface temperature profile was found
when temperatures were kept near the ambient temperature. When the temperature
reaches around 40 ◦ C, our solutions diverges from the experimental result, which is
much lower. The most challenging aspect of the problem currently is properly modeling the electromagnetic source in a fullwave solution. A gap current source failed
to track with the input power from the experiment, even when we tried to intervene
by forcing the current to increase when the conductivity increased.
68
The first recommendation for future work is an investigation into source modeling
for the fullwave RF ablation problem. In the static field problem, the source modeling
simple with the RF source being a voltage source and your unknown variable being a
scalar voltage. To solve a fullwave electromagnetic problem, a more complex source
with proper output impedance is necessary. Methods applied to similar problems
may be useful, or a new method could be designed. Alternatively, the electrostatic
approximation could be used to calculate the field distribution. This is the standard
method in the literature, and while the electromagnetic problem offers the ability
to simulate impedance matching, this benefit may not outweigh the added cost of
creating a fullwave model. Additionally, we recommend the use of convective boundary conditions in place of the insulating conditions. Their implementation should
be straightforward with potential benefits in modeling the surface temperature more
accurately. Another simple change to the thermal model would be to include the
temperature dependence of the product of density and specific heat, as discussed in
Chapter 4. Finally, a parametric study on how internal temperature manifests itself
in the surface temperature measured by IR thermography is needed. A study of this
nature would yield insight into how to correlate surface temperature measurements
with internal temperatures and could be carried out both numerically and in the lab.
69
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